Domain of 6/x: Determining Valid Input Values for a Fraction

Domain Restrictions with Division by Zero

Select the the domain of the following fraction:

6x \frac{6}{x}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the domain of substitution
00:03 Domain of substitution exists, to ensure we don't divide by 0
00:06 Meaning the denominator in the fraction must be different from 0
00:09 We will use this formula in our exercise
00:14 X must be different from 0, to avoid dividing by 0
00:17 Therefore this is the domain of substitution
00:20 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Select the the domain of the following fraction:

6x \frac{6}{x}

2

Step-by-step solution

The domain of a fraction depends on the denominator.

Since you cannot divide by zero, the denominator of a fraction cannot equal zero.

Therefore, for the fraction 6x \frac{6}{x} , the domain is "All numbers except 0," since the denominator cannot equal zero.

In other words, the domain is:

x0 x\ne0

3

Final Answer

All numbers except 0

Key Points to Remember

Essential concepts to master this topic
  • Division Rule: You cannot divide by zero in mathematics
  • Technique: Set denominator equal to zero: x = 0
  • Check: Domain excludes values that make denominator zero ✓

Common Mistakes

Avoid these frequent errors
  • Including zero in the domain
    Don't think all real numbers work for 6x \frac{6}{x} = undefined expression! When x = 0, you get 6/0 which is mathematically impossible. Always exclude values that make the denominator equal to zero.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

\( \frac{7}{7\cdot8}=8 \)

FAQ

Everything you need to know about this question

Why can't I divide by zero?

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Division by zero is undefined in mathematics! Think of it this way: if 6 ÷ 0 had an answer, then that answer times 0 would equal 6. But any number times 0 equals 0, not 6.

How do I find which values to exclude from the domain?

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Set the denominator equal to zero and solve! For 6x \frac{6}{x} , set x = 0. Since this is true when x = 0, exclude 0 from the domain.

What if the numerator is zero instead?

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That's totally fine! 0x \frac{0}{x} equals 0 (as long as x ≠ 0). The problem is only when the denominator is zero.

Can I substitute x = 0 to check my answer?

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No! Never substitute excluded values. Instead, try a value that is in the domain, like x = 1: 61=6 \frac{6}{1} = 6

What does 'all numbers except 0' mean in math notation?

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We write this as x0 x ≠ 0 or using interval notation: (-∞, 0) ∪ (0, ∞). This means all real numbers except zero.

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